1. ME 221Lecture 21 ME 221 Statics www.angel.msu.edu Sections 2.2 – 2.5

2. ME 221Lecture 22 Announcements ME221 TA’s and Help Sessions Chad Stimson – stimson1@msu.edu Homework grading & help room Tuesdays & Thursdays – 8am to 1pm – 1522EB Jimmy Issa – jimmy@msu.edu Quiz & exam grading & help room Tuesdays & Thursdays – 1pm to 5pm – 2415EB

3. ME 221Lecture 23 Announcements HW#1 Due on Friday, May 21 Chapter 1 - 1.1, 1.3, 1.4, 1.6, 1.7 Chapter 2 – 2.1, 2.2, 2.11, 2.15, 2.21 Quiz #1 on Friday, May 21

4. ME 221Lecture 24 Last Lecture Vectors, vectors, vectors Law of Cosines Law of Sines Drawing vector diagrams Chapter 1: Basics Example 1: Addition of Vectors

5. ME 221Lecture 25 Law of Cosines This will be used often in balancing forces c    b a

6. ME 221Lecture 26 Law of Sines Again, start with the same triangle    c b a

7. ME 221Lecture 27 300 lb 200 lb 45 o 25 o Example Note: resultant of two forces is the vectorial sum of the two vectors

8. ME 221Lecture 28 25 o 155 o 45 o   = 90 o +25 o -  200 lb 300 lb R 25 o 110 o 300 lb R 200 lb

9. ME 221Lecture 29 Scalar Multiplication of Vectors Multiplication of a vector by a scalar simply scales the magnitude with the direction unchanged Line of action A stays the same 0.5 x A

10. ME 221Lecture 210 Forces Review definition Shear and normal forces Resultant of coplanar forces

11. ME 221Lecture 211 Characteristics of a Force Its magnitude –denoted by |F| Its direction Its point of application –important when we discuss moments later

12. ME 221Lecture 212 Further Categorizing Forces Internal or external –external forces applied outside body P P A section of the body exposes internal body P Cut plane through body Internal tension

13. ME 221Lecture 213 Shear and Oblique Shear internal force has line of action contained in cutting plane P P Intenal shear forces

14. ME 221Lecture 214 Oblique Internal Forces Oblique cutting planes have both normal and shear components P S N Where N + S = P

15. ME 221Lecture 215 Transmissibility A force can be replaced by a force of equal magnitude provided it has the same line of action and does not disturb equilibrium B A

16. ME 221Lecture 216 Weight is a Force Weight is the force due to gravity –W = mg where m is mass and g is gravity constant g = 32.2 ft/s 2 = 9.81 m/s 2 English and metric –Weight lb or N –Mass slugs or kg

17. ME 221Lecture 217 Resultant of Coplanar Forces A body’s motion depends on the resultant of all the forces acting on it In 2-D, we can use the Laws of Sines and Cosines to determine the resultant force vector In 3-D, this is not practical and vector components must be utilized more on this later

18. ME 221Lecture 218 x y AxAx AyAy xx A yy yy Perpendicular Vectors A x is the component of vector A in the x-direction A y is the component of vector A in the y-direction AxAx xx AyAy x y A AyAy AxAx

19. ME 221Lecture 219 Vector Components Vector components are a powerful way to represent vectors in terms of coordinates. x y xx yy A where A x = |A| cos  x A y = |A| cos  y = |A| sin  x x y AxAx AyAy A= AxAyAxAy

20. ME 221Lecture 220 A x = |A| cos  x A y = |A| cos  y = |A| sin  x cos  x = A x / |A| cos  y = A y / |A| = x = y x and y are called direction cosines x 2 + y 2 = 1 Note: To apply this rule the two axes must be orthogonal Vector Components (continued)

21. ME 221Lecture 221 Summary External forces give rise to –tension and compression internal forces –normal and shear internal forces Forces can translate along their line of action without disturbing equilibrium The resultant force on a particle is the vector sum of the individual applied forces

22. ME 221Lecture 222 3-D Vectors; Base Vectors Rectangular Cartesian coordinates (3-D) Unit base vectors (2-D and 3-D) Arbitrary unit vectors Vector component manipulation

23. ME 221Lecture 223 3-D Rectangular Coordinates Coordinate axes are defined by Oxyz x y z O Coordinates can be rotated any way we like, but... Coordinate axes must be a right-handed coordinate system.

24. ME 221Lecture 224 x y z O A = Writing 3-D Components Component vectors add to give the vector: x y z O A = AxAx A x + AyAy A y + AzAz AzAz Also,

25. ME 221Lecture 225 3-D Direction Cosines The angle between the vector and coordinate axis measured in the plane of the two x y z O A xx yy zz Where: x 2 + y 2 + z 2 =1

26. ME 221Lecture 226 Unit Base Vectors Associate with each coordinate, x, y, and z, a unit vector (“hat”). All component calculations use the unit base vectors as grouping vectors. x y z O Now write vector as follows: where A x = |A x | A y = |A y | A z = |A z |

27. ME 221Lecture 227 Vector Equality in Components Two vectors are equal if they have equal components when referred to the same reference frame. That is: if A x = B x, A y = B y, A z = B z

28. ME 221Lecture 228 Additional Vector Operations To add vectors, simply group base vectors A scalar times vector A simply scales all the components

29. ME 221Lecture 229 General Unit Vectors Any vector divided by its magnitude forms a unit vector in the direction of the vector. –Again we use “hats” to designate unit vector x y z O b

30. ME 221Lecture 230 Position Vectors in Space Points A and B in space are referred to in terms of their position vectors. x y z O rArA rBrB r B/A Relative position defined by the difference A B

31. ME 221Lecture 231 Vectors in Matrix Form When using MatLab or setting up a system of equations, we will write vectors in a matrix form:

32. ME 221Lecture 232 Summary Write vector components in terms of base vectors Know how to generate a 3-D unit vector from any given vector

33. ME 221Lecture 233 Example Problem